Smallest Prime k-tuplets Oliver Atkin suggested that it would be of interest to record the smallest prime k-tuplet for each type of pattern. In many cases it is trivial, e.g. the two types of triplets are {5, 7, 11} and {7, 11, 13}. (Not {3, 5, 7} because all three residues of 3 are represented.) With larger k it is not so easy. For instance, the smallest 20-tuplet of the pattern {0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80} is clearly {29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109}. However, there are no known examples of the mirror-image pattern {0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80}. Contributions to this section are welcome. k=2 s=2 B={0 2} 3 5 5 7 11 13 17 19 29 31 41 43 59 61 71 73 101 103 107 109 k=3 s=6 B={0 4 6} 7 11 13 13 17 19 37 41 43 67 71 73 97 101 103 k=3 s=6 B={0 2 6} 5 7 11 11 13 17 17 19 23 41 43 47 101 103 107 k=4 s=8 B={0 2 6 8} 5 7 11 13 11 13 17 19 101 103 107 109 191 193 197 199 821 823 827 829 1481 1483 1487 1489 1871 1873 1877 1879 2081 2083 2087 2089 3251 3253 3257 3259 3461 3463 3467 3469 k=5 s=12 B={0 4 6 10 12} 7 11 13 17 19 97 101 103 107 109 1867 1871 1873 1877 1879 3457 3461 3463 3467 3469 5647 5651 5653 5657 5659 15727 15731 15733 15737 15739 k=5 s=12 B={0 2 6 8 12} 5 7 11 13 17 11 13 17 19 23 101 103 107 109 113 1481 1483 1487 1489 1493 k=6 s=16 B={0 4 6 10 12 16} 7 11 13 17 19 23 97 101 103 107 109 113 16057 16061 16063 16067 16069 16073 19417 19421 19423 19427 19429 19433 43777 43781 43783 43787 43789 43793 1091257 1091261 1091263 1091267 1091269 1091273 1615837 1615841 1615843 1615847 1615849 1615853 1954357 1954361 1954363 1954367 1954369 1954373 2822707 2822711 2822713 2822717 2822719 2822723 2839927 2839931 2839933 2839937 2839939 2839943 k=7 s=20 B={0 2 6 8 12 18 20} 11 13 17 19 23 29 31 165701 165703 165707 165709 165713 165719 165721 1068701 1068703 1068707 1068709 1068713 1068719 1068721 k=7 s=20 B={0 2 8 12 14 18 20} 5639 5641 5647 5651 5653 5657 5659 88799 88801 88807 88811 88813 88817 88819 284729 284731 284737 284741 284743 284747 284749 626609 626611 626617 626621 626623 626627 626629 855719 855721 855727 855731 855733 855737 855739 1146779 1146781 1146787 1146791 1146793 1146797 1146799 6560999 6561001 6561007 6561011 6561013 6561017 6561019 k=8 s=26 B={0 2 6 8 12 18 20 26} 11 13 17 19 23 29 31 37 k=8 s=26 B={0 2 6 12 14 20 24 26} 17 19 23 29 31 37 41 43 1277 1279 1283 1289 1291 1297 1301 1303 113147 113149 113153 113159 113161 113167 113171 113173 2580647 2580649 2580653 2580659 2580661 2580667 2580671 2580673 k=8 s=26 B={0 6 8 14 18 20 24 26} 88793 88799 88801 88807 88811 88813 88817 88819 284723 284729 284731 284737 284741 284743 284747 284749 855713 855719 855721 855727 855731 855733 855737 855739 1146773 1146779 1146781 1146787 1146791 1146793 1146797 1146799 6560993 6560999 6561001 6561007 6561011 6561013 6561017 6561019 Information for k = 9 provided by Vladimir Vlesycit k=9 s=30 B={0 4 6 10 16 18 24 28 30} Initial members of prime 9-tuplets: 13, 113143, 626927443, 2335215973, 3447123283, 4086982633, 4422726013, 6318867403, 7093284043, 8541306853, 10998082213, 14005112893, 18869466373, 21528117883, 21843411823, 28156779793, 30303283243, 31194463033, 33081664153, 35004115033, 35193551203 k=9 s=30 B={0 4 10 12 18 22 24 28 30} Initial members of prime 9-tuplets: 88789, 855709, 74266249, 964669609, 1422475909, 2117861719, 2558211559, 2873599429, 5766036949, 6568530949, 8076004609, 9853497739, 16394542249, 21171795079, 21956291869, 22741837819, 26486447149, 27254489389, 36955907689, 37045175329 k=9 s=30 B={0 2 6 8 12 18 20 26 30} Initial members of prime 9-tuplets: 11, 182403491, 226449521, 910935911, 1042090781, 1459270271, 2843348351, 6394117181, 6765896981, 8247812381, 8750853101, 11076719651, 12850665671, 17140322651, 22784826131, 24816950771, 33081664151, 41614070411, 43298074271, 43813839521, 53001578081, 54270148391, 57440594201 k=9 s=30 B={0 2 6 12 14 20 24 26 30} Initial members of prime 9-tuplets: 17, 1277, 113147, 252277007, 408936947, 521481197, 1116452627, 1209950867, 1645175087, 2966003057, 3947480417, 6234613727, 9307040837, 9853497737, 11878692167, 13766391467, 21956291867, 22741837817, 24388061207, 24718113437, 28498194707 k=10 s=32: Provided by Warut Roonguthai: Initial members of prime decuplets (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26, p+30, p+32). 11, 33081664151, 83122625471, 294920291201, 573459229151, 663903555851, 688697679401, 730121110331, 1044815397161, 1089869189021, 1108671297731, 1235039237891, 1291458592421, 1738278660731, 1761428046221, 1769102630411, 1804037746781, 1944541048181, 2135766611591, 2177961410741, 2206701370871 Initial members of prime decuplets (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26, p+30, p+32). 9853497737, 21956291867, 22741837817, 164444511587, 179590045487, 217999764107, 231255798857, 242360943257, 666413245007, 696391309697, 867132039857, 974275568237, 976136848847, 1002263588297, 1086344116367, 1403337211247, 1418575498577, 2118274828907, 2202874566017, 2385504217067 Information for k = 11 provided by Vladimir Vlesycit k=11 s=36 B={0 4 6 10 16 18 24 28 30 34 36} Initial members of prime 11-tuplets: 1418575498573, 2118274828903, 4396774576273, 6368171154193, 6953798916913, 27899359258003, 28138953913303, 34460918582323, 40362095929003, 42023308245613, 44058461657443, 61062361183903, 76075560855373, 80114623697803, 84510447435493, 85160397055813, 90589658803723, 90793299817453, 147119918235523, 156393789335863 k=11 s=36 B={0 2 6 8 12 18 20 26 30 32 36} Initial members of prime k-tuplets: 11, 7908189600581, 10527733922591, 12640876669691, 38545620633251, 43564522846961, 60268613366231, 60596839933361, 71431649320301, 79405799458871, 109319665100531, 153467532929981, 171316998238271, 216585060731771, 254583955361621, 259685796605351, 268349524548221, 290857309443971, 295606138063121, 380284918609481, 437163765888581 Information for k = 13 provided by Vladimir Vlesycit k=12 s=42 B={0 6 10 12 16 22 24 30 34 36 40 42} Initial members of prime 12-tuplets: 1418575498567, 27899359257997, 34460918582317, 76075560855367, 186460616596327, 218021188549237, 234280497145537, 282854319391717, 345120905374087, 346117552180627, 604439135284057, 727417501795057, 1041814617748747, 1090754719898917, 1539765965257747, 3152045700948217, 3323127757029307, 3449427485143867, 4422879865247917, 4525595253333997 k=12 s=42 B={0 2 6 8 12 18 20 26 30 32 36 42} Initial members of prime 12-tuplets: 11, 380284918609481, 437163765888581, 701889794782061, 980125031081081, 1277156391416021, 1487854607298791, 1833994713165731, 2115067287743141, 2325810733931801, 3056805353932061, 3252606350489381, 3360877662097841, 3501482688249431, 3595802556731501, 3843547642594391, 5000014653723821, 5861268883004651, 7486645325734691, 7933248530182091, 8760935349271991 Information for k = 13 provided by Vladimir Vlesycit k=13 s=48 B={0 6 12 16 18 22 28 30 36 40 42 46 48} Initial members of prime 13-tuplets: 186460616596321, 7582919852522851, 31979851757518501, 49357906247864281, 79287805466244211, 85276506263432551, 89309633704415191, 89374633724310001, 98147762882334001, 136667406812471371, 137803293675931951, 152004604862224951, 157168285586497021, 159054409963103491, 191495873018073691, 210509800417460611, 301056055256880991, 379374319296406501, 388218745698928981, 389259253111633741 k=13 s=48 B={0 4 6 10 16 18 24 28 30 34 40 46 48} Initial members of prime 13-tuplets: 13, 4289907938811613, 5693002600430263, 21817283854511263, 48290946353555023, 51165618791484133, 53094081535451893, 70219878257874463, 98633358468021313, 99142644093930373, 104814760374339133, 166784569423739203, 167841416726358493, 184601252515266523, 263331429949004353, 272039012072134243, 339094624362619243, 363319822006646623, 363760043662280383, 437335541550793003, 455289126169953193 k=13 s=48 B={0 4 6 10 16 18 24 28 30 34 36 46 48} Initial members of prime 13-tuplets: 1707898733581273, 3266590043460823, 4422879865247923, 10907318641689703, 32472302129057023, 52590359764282573, 60229684381540753, 67893346321234513, 93179596929433093, 115458868925574253, 140563537593599353, 142977538681261363, 148877505784397623, 166362638531783773, 232442516762530153, 236585787518684683, 255933372890105143, 317294052871840123, 325853825645632363, 337188071215909993, 344447962857168403 k=13 s=48 B={0 2 6 8 12 18 20 26 30 32 36 42 48} Initial members of prime 13-tuplets: 11, 7933248530182091, 20475715985020181, 21817283854511261, 33502273017038711, 40257009922154141, 49242777550551701, 49600456951571411, 75093141517810301, 84653373093824651, 119308586807395871, 129037438367672951, 129706953139869221, 151242381725881331, 158947009165390331, 161216594737343261, 167317340088093311, 176587730173540571, 178444395317213141, 197053322268438521, 301854920123441801 k=13 s=48 B={0 2 8 14 18 20 24 30 32 38 42 44 48} Initial members of prime 13-tuplets: 7697168877290909, 10071192314217869, 11987120084474369, 28561589689237439, 62321320746357689, 73698766709402669, 75046774774314359, 79287805466244209, 98551408299919409, 136720189890477209, 225735856757596019, 234065221633327919, 302834818301440259, 360345440708336099, 385443070970192069, 387494664213890249, 466256026285842809, 539043082132918379, 570108181108560929, 610147978081735109 k=13 s=48 B={0 2 12 14 18 20 24 30 32 38 42 44 48} Initial members of prime 13-tuplets: 10527733922579, 15991086371740199, 22443709342850669, 69759046409087909, 94415460183744419, 164873121596539229, 197053322268438509, 212971209388223159, 215768926871613989, 248170682800139819, 270109976153617319, 326374793491266239, 341896216415143109, 341987213500572359, 362035072661912369, 401062754451879239, 441180406661470349, 450928996714672349, 503035098004929209, 533306698691196149 k=14 s=50 B={0 2 6 8 12 18 20 26 30 32 36 42 48 50} 11 13 17 19 23 29 31 37 41 43 47 53 59 61 21817283854511261 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (1982, D. Betsis & S. Säfholm) These initial members of prime 14-tuplets were provided by Vladimir Vlesycit: 841262446570150721, 1006587882969594041, 2682372491413700201, 5009128141636113611, 7126352574372296381, 7993822923596334941, 12870536149631655611, 15762479629138737611 k=14 s=50 B={0 2 8 14 18 20 24 30 32 38 42 44 48 50} 79287805466244209 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (1982, D. Betsis & S. Säfholm) These initial members of prime 14-tuplets were provided by Vladimir Vlesycit: 2714623996387988519, 5012524663381750349, 6120794469172998449, 6195991854028811669, 6232932509314786109, 6808488664768715759, 10756418345074847279, 11319107721272355839, 12635619305675250719, 14028155447337025829, 14094050870111867489, 14603617704434643719, 14777669568340323479, 15420967329931107779, 16222575536498135639, 16624441191356313149, 17367037621075657349 k=15 s=56 B={0 2 6 8 12 18 20 26 30 32 36 42 48 50 56} 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 44360646117391789301 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (20 digits, 2001, Tom Hadley) k=15 s=56 B={0 2 6 12 14 20 24 26 30 36 42 44 50 54 56} 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 17905159760365247387 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (20 digits, 2001, Jim Morton) k=15 s=56 B={0 2 6 12 14 20 26 30 32 36 42 44 50 54 56} 1240068005144831867 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (1999, Joerg Waldvogel) k=15 s=56 B={0 6 8 14 20 24 26 30 36 38 44 48 50 54 56} 348214184662549960583 + 0, 6, 8, 14, 20, 24, 26, 30, 36, 38, 44, 48, 50, 54, 56 (1999, Joerg Waldvogel) k=16 s=60 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60} 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 k=16 s=60 B={0 2 6 12 14 20 26 30 32 36 42 44 50 54 56 60} 47710850533373130107 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (1997, TF and Joerg Waldvogel) k=17 s=66 B={0 4 10 12 16 22 24 30 36 40 42 46 52 54 60 64 66} None in the interval 0 to (1/3)*10^24 (Peter Leikauf and Joerg Waldvogel) k=17 s=66 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60 66} Initial members of prime 17-tuplets provided by Peter Leikauf and Joerg Waldvogel: 13, 47624415490498763963983, 78314167738064529047713, 83405687980406998933663, 110885131130067570042703, 163027495131423420474913 k=17 s=66 B={0 6 8 12 18 20 26 32 36 38 42 48 50 56 60 62 66} 1620784518619319025971 + 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 (1997, Joerg Waldvogel) Initial members of prime 17-tuplets provided by Peter Leikauf and Joerg Waldvogel: 1620784518619319025971, 2639154464612254121531, 3259125690557440336631, 124211857692162527019731 k=17 s=66 B={0 2 6 12 14 20 24 26 30 36 42 44 50 54 56 62 66} Initial members of prime 17-tuplets provided by Peter Leikauf and Joerg Waldvogel: 17, 37630850994954402655487, 53947453971035573715707, 174856263959258260646207, 176964638100452596444067, 207068890313310815346497, 247620555224812786876877, 322237784423505559739147 k=18 s=70 B={0 4 10 12 16 22 24 30 36 40 42 46 52 54 60 64 66 70} 2845372542509911868266807 + 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (2000, Joerg Waldvogel & Peter Leikauf) k=18 s=70 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60 66 70} 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 1906230835046648293290043 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70 (2001, Joerg Waldvogel & Peter Leikauf) k=19 s=76 B={0 6 10 16 18 22 28 30 36 42 46 48 52 58 60 66 70 72 76} No known examples of this pattern k=19 s=76 B={0 4 6 10 16 22 24 30 34 36 42 46 52 60 64 66 70 72 76} 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=19 s=76 B={0 4 6 10 12 16 24 30 34 40 42 46 52 54 60 66 70 72 76} No known examples of this pattern k=19 s=76 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60 66 70 76} 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 k=20 s=80 B={0 2 6 8 12 20 26 30 36 38 42 48 50 56 62 66 68 72 78 80} 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 k=20 s=80 B={0 2 8 12 14 18 24 30 32 38 42 44 50 54 60 68 72 74 78 80} No known examples of this pattern Known prime k-tuplets for 21 <= k <= 87: k=21 s=84 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=22 s=90 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 k=22 s=90 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=23 s=94 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=25 s=110 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 k=27 s=120 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 k=33 s=152 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 k=38 s=176 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 k=50 s=246 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 k=51 s=252 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 k=51 s=252 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 k=52 s=254 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 k=53 s=264 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 k=77 s=420 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 k=77 s=420 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 k=77 s=420 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 k=78 s=422 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 Last updated: 21 November 2001